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Yes, you can play A Aeolian over Am, C Ionian over C, E Phrygian over Em and F Lydian over F. That would mean in fact that you'll be using the exact same notes for all of them! Because they all come from the same parent key (C Ionian). Except you may consider the note names themselves as 'home'.

Why you'd use E Aeolian over an Em chord is a good point. Doing this changes the F you may play into F#. That's not diatonic to the key you're in, although it may (or may not) give a little extra spice to the solo, depending where and how it emerges.

Same sort of concept with F Ionian over the F chord. That'll give a Bb note instead of a B nat. Again, it may (or may not) fit well into what's being played at that point. I'd say that if the F chord followed the C, then F Ionian may work slightly better.

If you are looking for a 'rule' to work to, there isn't one, as it's theory. But, as always, let your ears be the deciding judges. At least with the snippets contained in this answer, you have some options!

I often say to students "At some point, you'll be able to play any note anywhere over any chord." It's just a matter of being aware of what might work, and what most likely won't work. Timing, dynamics, note placing in the bar, note value, all come into the equation. As an example, try playing as quavers, B C D# E F# G B C D# E. Sounds fine over a simple C chord - but half the notes are (theoretically) wrong!